1. Field of the Invention
The present invention relates to a free-form surface creation method and apparatus thereof and is applicable to create geometries with a free-form surface by, for example, a CAD (computer-aided design) or CAM (computer-aided manufacturing) system.
2. Description of the Related Art
In the case of designing the geometry of a body with a free-form surface by using, for example, CAD technique (geometric modeling), a designer generally specifies multiple points in a three-dimensional space (which are called "nodes"), and the boundary curve network connecting respective specified multiple nodes is calculated by a computer using a predetermined vector function, thereby creating a surface represented in wire frames. Thus, it is possible to create a number of frame spaces delimited by boundary curves (hereafter, such processing is referred to as "frame processing").
The boundary curve network created by such frame processing represents by itself a general form that the designer wants to design. If it is possible to interpolating calculate a surface which can be represented by the predetermined vector function by using the boundary curves delimiting each frame space, a free-form surface (which cannot be defined by a quadratic function) designed by the designer can be created as a whole. Here, the free-form surfaces patched in each frame space form the basic elements constituting the whole surface, which are called patches.
Conventionally, in this type of CAD system, as the vector function representing the boundary curve network, cubic tensor products consisting of, for example, a Bezier expression or B-spline expression have been used for ease of calculation, which are considered to be optimal for mathematical representation of such free-form surfaces that have no special geometrical feature. That is, in a free-form surface having no special geometrical feature, when given points in the space are projected onto the x-y plane, the respective projected points are often regularly arranged in the form of a matrix. And it is known that, when the number of these projected points is represented by m.times.n, the respective frame spaces can be easily patched with a quadrilateral patch represented by a cubic Bezier expression. However, this conventional mathematical representation has a difficulty in the connection of each patch if it is applied to a surface having special geometrical features (for example, a surface with a greatly distorted geometry). And, this involves advanced mathematical operations so that the processing by computer is huge and complex, thus there occurs a problem that a long operation time is required.
As disclosed in Japanese laid open patent applications 62-135965, 62-151978, 62-157968, 62-173569, 62-173569, 62-190564, 62-216076, 62-221073, 62-224863, and 62-226281, as a method to solve the problem has been proposed, which obtains such internal control points that satisfy the condition of tangent continuation at the common boundary between adjacent frame spaces and use the vector function representing the free-form surface determined by the respective internal control points to patch the patches of free-form surface.
As shown in FIG. 1, in computer-aided artistic design involving free-form surfaces, a smooth surface is represented in a combination of several patches. Generally, in such a case, only patch boundaries are defined as the frames and no internal geometry is defined. Smoothness is the only requirement in most cases. Here, as the methods to determine internal control point vectors P.sub.11, P.sub.12, P.sub.21 and P.sub.22 defining the internal geometry of a bi-cubic Bezier patch given its boundaries shown in FIG. 2, there have been the methods such that the twist vectors at the four corners of the patch are set to zero, or such that the second-order differential coefficients in the "u" and "v" directions are set to zero, as disclosed by a Japanese Laid-Open Patent Application 62-221073 and pages 65-66, Journal of Precise Engineering Association of Japan 56/3/1990.
However, determining internal control points by the method of setting twist vectors to zero causes the internal geometry of the patch to be-greatly influenced by the frame that forms the boundary of that patch. The method of setting the second-order differential coefficients to zero exerts less influence by the frame than the method of setting the twist vectors to zero, however, it has the feature of giving a flat impression. Both methods aim principally at the smoothness in the internal geometry of patches, so that there occurs a problem of no consideration given for connection to adjacent patches.
As shown in FIG. 3, for connecting two adjacent patches A and B, it is known that it is possible to connect the tangent of the patches A and B successively by making the patches A and B to share an edge "b" and making the respective normal lines at the point vectors "f" and "g" on the patches A and B to coincide with each other, as disclosed by a Japanese Laid-Open Patent Application 01-175672. However, in FIG. 4, for connection of the patch A and patch B, internal control point vectors "a", "b", "c" and "d" are to be adjusted, and for connection of the patch B and patch C, internal control point vectors "d", "e", "f" and "g" are to be adjusted. Thus, the internal control point vector "d" is required for connection of both the patches A and B and the patches B and C, so that it is necessary to determine the internal control point vector "d" with consideration for connection of both the patches A and B and the patches B and C. However, this operation has been performed manually, and it is difficult and takes much labor to connect all of the patch A, patch B, patch C and patch D.